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Many practical optimization problems involve nonsmooth functions. In this chapter, we give an extensive collection of problems for nonsmooth minimization which can be used to test nonsmooth optimization solvers. The general formula for these problems is written by

$$\begin{aligned} {\left\{ \begin{array}{ll} \text {minimize}\quad f({{\varvec{x}}}) &{}\\ \text {subject to} \quad {\varvec{x}}\in S,&{} \end{array}\right. } \end{aligned}$$
(9.1)

where the objective function \(f:\mathbb {R}^n \rightarrow \mathbb {R}\) is supposed to be locally Lipschitz continuous on the feasible region \(S \subseteq \mathbb {R}^n\). Note that no differentiability or convexity assumptions are made. All the problems given here are found in the literature and have been used in the past to develop, test, or compare NSO software.

We shall use a classification of test problems modified from that of [111]. That is we use a sequence of letters

$$\begin{aligned} \text {O[O]-C-R-S}, \end{aligned}$$

where Table 9.1 give all possible abbreviations that could replace the letters O, C, R, and S (the brackets mean an optional abbreviation).

We first give a summary of all the problems in Table 9.2, where \(n\) denotes the number of variables and \(f({\varvec{x}}^*)\) is the minimum value of the objective function. In addition, the classification and the references to the origin of the problems in each case are given in Table 9.2. Then, in Sects. 9.1 (small unconstrained problems), 9.2 (bound constraints), 9.3 (linearly constrained problems), 9.4 (large-scale unconstrained problems), and 9.5 (inequality constraints), we present the formulation of the objective function \(f\), possible constraint functions \(g_j\) (\(j=1,\ldots ,p\)), and the starting point \({\varvec{x}}^{(1)}=(x_1^{(1)},\ldots ,x_n^{(1)})^T\) for each problem. We also give the minimum point \({\varvec{x}}^*=(x_1^*,\ldots ,x_n^*)^T\) for the problems with the precision of at least four decimals if possible (in larger cases this is not always practicable).

In what follows, we denote by \({{\mathrm{div}}}\,(i,j)\) the integer division for positive integers \(i\) and \(j\), that is, the maximum integer not greater than \(i/j\), and by \({{\mathrm{mod}}}\,(i,j)\) the remainder after integer division, that is, \({{\mathrm{mod}}}\,(i,j)=j(i/j-{{\mathrm{div}}}\,(i,j))\).

Table 9.1 Classification of test problems
Full size table
Table 9.2 Condensed list of all test problems
Full size table
Table 9.3 Values of vectors \({\varvec{s}}\) and \({\varvec{d}}\) for problem 19
Full size table
Table 9.4 Data for symmetric cost matrix A for problem 19 (cont.)
Full size table

1 Small Unconstrained Problems

In this section we describe 40 small-scale nonsmooth unconstrained test problems. The number of variables varies from 2 to 50.

1

CB2 (Charalambous/Bandler) [59]

figure a

2

CB3 (Charalambous/Bandler) [59]

figure b

3

DEM [168]

figure c

4

QL [168]

figure d

5

LQ [168]

figure e

6

Mifflin 1 [168]

figure f

7

Wolfe [159]

figure g

8

Rosen-Suzuki [205]

figure h

9

Davidon 2 [159]

figure i

10

Shor [168]

figure j

11

Maxquad [146,168]

figure k

12

Polak 2 [198]

figure l

13

Polak 3 [198]

figure m

14

Wong 1 [7,159]

figure n

15

Wong 2 [7,159]

figure o

16

Wong 3 [7,159]

figure p

17

MAXQ [208]

figure q

18

MAXL [168]

figure r
Table 9.5 Initialization and optimal point for 19
Full size table

19

TR48 [146]

figure s

20

Goffin [168]

figure t

21

Crescent [131]

figure u

22

Mifflin 2 [168]

figure v

23

WF [159]

figure w

24

SPIRAL [159]

figure x

25

EVD52 [159]

figure y

26

PBC3 [159]

figure z

27

Bard [159]

figure aa

28

Kowalik-Osborne [159]

figure ab

29

Polak 6 [198]

figure ac

30

OET5 [159]

figure ad

31

OET6 [159]

figure ae

32

EXP [159]

figure af

33

PBC1 [159]

figure ag

34

HS78 [111,159]

figure ah

35

El-Attar [159]

figure ai

36

EVD61 [159]

figure aj

37

Gill [159]

figure ak

38

Problem 1 in [21]

figure al

39

L1 version of Rosenbrock function [21]

figure am

40

L1 version of Wood function [21]

figure an

2 Bound Constrained Problems

Bound constrained problems (\(S = \{{\varvec{x}}\in \mathbb {R}^n \mid x_i^l \le x_i \le x_i^u \text { for all }i=1,\ldots ,n \}\) in (9.1)) are easily constructed from the problems given above, for instance, by inclosing the bounds

$$\begin{aligned} x^*_i + 0.1 \le x_i \le x^*_i + 1.1 \qquad \text {for all odd } i. \end{aligned}$$

If the starting point \({\varvec{x}}^{(1)}\) is not feasible, it can simply be projected to the feasible region (if a strictly feasible starting point is needed an additional safeguard of 0.0001 may be added). The classification of the bound constrained problems is the same as that of unconstrained problems (see Sect. 9.1 and also Sect. 9.4) but, naturally, the information about constraint functions should be replaced with B (see Table 9.1).

3 Linearly Constrained Problems

In this section we present small-scale nonsmooth linearly constrained test problems (\(S = \{{\varvec{x}}\in \mathbb {R}^n \mid A {\varvec{x}}\le {\varvec{b}}\}\) with the inequality taken component-wise in (9.1)). The number of variables varies from 2 to 20 and there are up to 15 constraint functions.

41

Wong 2C [159]

figure ao

42

Wong 3C [159]

figure ap

43

MAD1 [159]

figure aq

44

MAD2 [159]

figure ar

45

MAD4 [159]

figure as

46

MAD5 [159]

figure at

47

PENTAGON [159]

figure au

48

MAD6 [159]

figure av

49

Dembo 3 [159]

figure aw

50

Dembo 5 [159]

figure ax

51

EQUIL [146,159]

figure ay

52

HS114 [159]

figure az

53

Dembo 7 [159]

figure ba

54

MAD8 [159]

figure bb
Table 9.6 Values of \({\varvec{a}}\) for problem 49
Full size table

4 Large Problems

In this section we present 21 large-scale nonsmooth unconstrained test problems. The problems can be formulated with any number of variables.

55

Generalization of MAXL [155]

figure bc

56

Generalization of L1HILB [155]

figure bd

57

Generalization of MAXQ [98]

figure be

58

Generalization of MXHILB [98]

figure bf

59

Chained LQ [98]

figure bg

60

Chained CB3 I [98]

figure bh

61

Chained CB3 II [98]

figure bi

62

Number of active faces [95]

figure bj

63

Nonsmooth generalization of Brown function 2 [98]

figure bk

64

Chained Mifflin 2 [98]

figure bl

65

Chained crescent I [98]

figure bm

66

Chained crescent II [98]

figure bn

67

Problem 6 in TEST29 [155]

figure bo

68

Problem 17 in TEST29 [155]

figure bp

69

Problem 19 in TEST29 [155]

figure bq

70

Problem 20 in TEST29 [155]

figure br

71

Problem 22 in TEST29 [155]

figure bs

72

Problem 24 in TEST29 [155]

figure bt

73

DC Maxl [21]

figure bu

74

DC Maxq [26]

figure bv

75

Problem 6 in [26]

figure bw

76

Problem 7 in [26]

figure bx

Similarly to small-scale problems these problems can be turned to bound constrained ones, for instance, by inclosing the additional bounds

$$\begin{aligned} x^*_i + 0.1 \le x_i \le x^*_i + 1.1 \qquad \text {for all odd } i. \end{aligned}$$

5 Inequality Constrained Problems

In this section, we describe five nonlinear or nonsmooth inequality constraints (or constraint combinations). The constraints can be combined with the problems 57–66 given in Sect. 9.4 to obtain 50 inequality constrained problems (\(S = \{{\varvec{x}}\in \mathbb {R}^n \mid g_j({\varvec{x}})\le 0 \text { for all }j=1,\ldots ,p \}\) in (9.1)).

The constraints are selected such that the original unconstrained minimizers of problems in Sect. 9.4 are not feasible. Note that, due to nonconvexity of the constraints, all the inequality constrained problems formed this way are nonconvex.

The starting points \({\varvec{x}}^{(1)}=(x_1^{(1)},\ldots ,x_n^{(1)})^T\) for inequality constrained problems are chosen to be strictly feasible. In what follows, the starting points for the problems with constraints are the same as those for problems without constraints (see Sect. 9.4) unless stated otherwise. The optimum values for the problems with different objective functions and \(n=1000\) are given in Table 9.2.

77

Modification of Broyden tridiagonal constraint I [122,126]

figure by

78

Modification of Broyden tridiagonal constraint II [122,126]

figure bz

79

Modification of MAD1 I [122, 126]

figure ca

80

Modification of MAD1 II [122, 126]

figure cb

81

Simple modification of MAD1 [122,126]

figure cc